The direct conversion of electrical energy to mechanical work by a material is relevant to a number of applications. This is illustrated by ferroelectric 'relaxors' such as Pb(Mg(1/3)Nb(2/3))O(3)-PbTiO(3) (PMN-PT; refs 5, 6): these materials exhibit a giant electromechanical (piezoelectric) response that is finding use in ultrasonic and medical applications, as well as in telecommunications. The origins of this effect are, however, still unclear. Here we show that the giant electromechanical response in PMN-PT (and potentially other ferroelectric relaxors) is the manifestation of critical points that define a line in the phase diagram of this system. Specifically, in the electric-field-temperature-composition phase diagram of PMN-PT (the composition being varied by changing the PT concentration), a first-order paraelectric-ferroelectric phase transition terminates in a line of critical points where the piezoelectric coefficient is maximum. Above this line, supercritical evolution is observed. On approaching the critical point, both the energy cost and the electric field necessary to induce ferroelectric polarization rotations decrease significantly, thus explaining the giant electromechanical response of these relaxors.